Is there a burgeoning Republican Naturist movement, of which till now we have not been aware?

Perhaps it is a new development in along the theme of “family values”? E.g., from the AANR FAQ:

Children are natural nudists and quickly take to it like ducks take to water. Most clubs welcome children and encourage families to join. At these clubs, you’ll find activities and facilities for your kids to enjoy - and they’ll thank you for taking them along. As always, we recommend that if you are bringing your children, you call ahead to ask the club about its activity program and facilities. For more information please see our discussion of family values.

See their Family Values page.

Republicans and nudists naturists, natural political allies?